# To find the equation of a space curve given a curvature and torsion function in terms of the angle between the Darboux vector and the tangent

If a space curve $\gamma(s)$ parametrised with respect to arc length $s$ has curvature as $$\kappa(s) = \cos^{2} (\theta(s))$$ and torsion as $$\tau(s) = \sin(\theta(s) ) \cos(\theta(s)) ,$$ where $\theta(s)$ is a function that gives the angle between the tangent vector to the curve and it's Darboux vector i.e. $\tau T + \kappa B$ , then is there a way of getting the possible parametrisation of such a curve or even $\textbf{an example of such a curve}$??

Of course, apart from the circular helix, in which case the function $\theta$ is a constant.

$\textbf{Note}$ : The angle $\theta(s)$ is actually the angle between the tangent to $\gamma$ and the tangent vector of another curve which is $1-1$ correspondence with $\gamma$, however that tangent vector is itself parallel to the Darboux vector, so I have defined it as such.

• Do you mean a curve for which $\kappa$ and $\tau$ satisfy the given conditions? If so, you shouldn't expect many solutions: We can write $\cos^2 \theta$ and $\sin \theta \cos \theta$ as functions of $\kappa$ and $\theta$, giving a system of two equations in two unknowns, which generically has finite many solutions. In that case, $\kappa$ and $\tau$ must be locally constant, so one expects the only solutions to be certain helices. – Travis Jun 9 '17 at 12:36
• @Travis Yes. I did mean the same. But I didnt get why they would be helices as that would force $\theta$ to be a constant, right?? By locally constant, did you mean, like a curve patched together by helices of different radius? – Vishesh Jun 9 '17 at 16:33
• So, if one carries out the computation I described, you get the solutions $\kappa = \frac{1}{2}$, $\tau = \pm \frac{1}{2}$. But the only curves with constant curvature and torsion are helices. In fact, that this is the solution to the system shows that there's only a single solution up to Euclidean motions (allowing reflections). – Travis Jun 9 '17 at 16:36
• @Travis. Well the curvature and torsion functions seem to satisfy a relation like $\kappa^2 +\tau^2 = \kappa$ and I could think of non-helical curves too which satisfy that relation. So I presumed there may be other curves than the helix. Also I didn't get as to how you ended with those values. The varying angle condition seems to be vacuously satisfied then. But the fundamental theorem seems to hint that s not the case, right? – Vishesh Jun 9 '17 at 16:59